Moscow Mathematical Journal

Volume 14, Issue 3, July–September 2014 pp. 617–637.

On Point-Like Interaction of Three Particles: Two Fermions and Another Particle. II

Authors: R.A. Minlos (1)
Author institution:(1) Institute for Information Transmission Problems of Russian Academy of Sciences, Bolshoy Karetnyi 19, Moscow, Russia

Summary:

This work continues our previous article, where the construction of Hamiltonian H for the system of three quantum particles is considered. Namely the system consists of two fermions with mass 1 and another particle with mass m>0. In the present paper, like before, we study the part T_l=1 of auxilliary operator T = ⊕^∞_l=0 T_l involving the construction of the resolvent for the operator H. In this work together with the previous one two constants 0<m₁<m₀<∞ were found such that: 1) for m>m₀ the operator T_l=1 is selfadjoint but for m≤m₀ it has the deficiency indexes (1, 1); 2) for m₁<m<m₀ any selfadjoint extension of T_l=1 is semibounded from below; 3) for 0<m<m₁ any selfadjoint extension of T_l=1 has the sequence of eigenvalues {λ_n<0, n>n₀} with the asymptotics λ_n = λ₀ e^δn + O(1), n → ∞, where λ₀<0, δ > 0, n₀>0 and there is no other spectrum on the interval λ<λ_n₀.

2010 Mathematics Subject Classification. 81Q10, 47S30.

Keywords: Selfadjoint extension, Mellin’s transformation, formula of Sokhotsky, boundedness from below, deficincy index.

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